A floor response spectrum is a type of design response spectrum developed for a certain location in a primary structure. The primary structure will, through its natural frequencies, act as a bandpass filter for the original signal. Thus, the floor response spectrum will typically have significant peaks related to the natural frequencies of the primary structure. The term floor response spectrum is derived from the fact that this local response spectrum will typically be different between different floors of a building.
A large system, like a piping system, may not have the same floor response spectrum at all of its support points. This causes significant complications to the analysis. Assume that a mathematical model of a structure is discretized by FEM so that the equations of motion on matrix form are.
The structure is at a number of points connected to a common "ground" that has the base motion. The relative displacement is now. With no external load, the equation of motion is. Here, the fact that a rigid body motion does not introduce any elastic or viscous forces in the system has been used, so that. By solving the undamped eigenvalue problem with the support nodes being fixed, a set of N eigenmodes.
These eigenmodes can represent the relative displacements but not the absolute displacements , since all eigenmodes will have zero displacements at the support points in an eigenfrequency analysis. By standard operations for mode superposition , the decoupled modal equations are. It has been assumed that the mass matrix normalization of the eigenmodes is used and that the damping matrix can be diagonalized by the eigenmodes. The mass matrix normalization is not essential, but it will simplify certain expressions.
Here, is the modal coefficient for mode j , so that the relative displacement can be written as a linear combination of eigenmodes, weighted by the modal coordinates:. The vector has the value "1" in all X -translation DOFs and the value "0" in all other.
The modal equation of motion is then. The multipliers are the modal participation factors ;. Thus, the maximum amplitude of mode j , when loaded by a base motion in direction k described by a response spectrum, is. To summarize, the peak amplitude for a certain eigenmode is the product of the response spectrum value at the corresponding natural frequency which is independent of the structure and the participation factor which is a property of the structure but independent of the loading.
In practice, several modes will have natural frequencies in the frequency range covered by the design response spectrum. This means that some combination of their responses is needed. There are several rules for how this combination can be arranged, as will be described in detail below.
These summation rules are nonlinear. For all combination types, all result quantities are strictly positive. As an effect, any quantity must be summed based on its own modal response. For example, stress components must be computed using the modal stresses and cannot be recovered from the summed strains, and strains cannot be recovered from summed displacements.
This has many consequences for the interpretation of results from a response spectrum analysis. Some examples are:. Often, the excitation is given in three orthogonal directions. The general approach is to consider the excitation in the three directions separately. First, all modal responses are summed for each direction, and then the results for the three directions are summed. An exception is the CQC3 summation rule, described below, in which the spatial and modal summation is done at the same time.
It is often useful to divide the eigenmodes into periodic modes and rigid modes. The distinction is related to the frequency content of the excitation relative to the eigenfrequency of the mode. In a high-frequency mode, the mass of the oscillator will mainly be translated in phase with the support. Such modes constitute the rigid modes. Their responses are synchronous with each other and with the base motion. This means that for rigid modes, a pure summation including signs should be used.
Modes with a significant dynamic response constitute the periodic modes. The maximum values for such modes will be more or less randomly distributed in time, since their periods differ. For this reason, the periodic part of the response requires more sophisticated summation techniques. A plain summation of the maximum values will, in general, significantly overestimate the true response. Modes that are in a transition region will partially contribute to the periodic modes and partially to the rigid ones.
In addition, it is sometimes necessary to add some static load cases containing a missing mass correction. Not all analyses require a separation into periodic and rigid modes. If not, all modes are treated as periodic. In the following, denotes any result quantity caused by excitation in direction. The periodic part of is denoted , and the rigid part is denoted.
Similarly, and denote the results from an individual eigenmode j. There are two different methods in use by which partitioning can be done. In either case, for mode j ,.
The difference between the two methods lies in how the coefficient is determined. For low frequencies, it should approach the value 0, and for high frequencies, the value 1. In the Gupta method, is a linear function of the logarithm of the natural frequency.
Here, and are two key frequencies. Thus, for eigenfrequencies below , the modes are considered as purely periodic, and above , purely rigid. In the original Gupta method, the lower key frequency is given by.
Here, and are the maximum values of the acceleration and velocity spectra, respectively. In the idealized spectrum above, this occurs at the point D. The second key frequency should be chosen so that the modes above this frequency behave as rigid modes.
The frequency can be taken as the one where the response spectra for different damping ratios converge to each other. In the Lindley-Yow method, the coefficient depends directly on the response spectrum values, not only on the frequency. As a consequence, it is possible that a certain mode can be considered as having a different degree of rigidness for different excitation directions.
The so-called zero period acceleration ZPA is the maximum ground acceleration during the event,. This is also the high-frequency asymptotic value of the absolute acceleration or pseudoacceleration in the response spectrum.
It corresponds to the F-G part of the idealized spectrum. The value of must, for physical reasons, be in the range of 0 to 1 and increase with frequency. Once the periodic and rigid responses for all modes have been summed up separately, they are combined as.
The most conservative method is to sum the maximum response for all N modes, thus assuming that all modes reach their maximum at the same time. In many cases, this approach leads to a design that is significantly overconservative. In the worst case scenario, the predicted result using N , not closely spaced modes can be a factor larger than what would be obtained using the other methods below.
The most popular method for superposition of the periodic modes is the complete quadratic combination CQC method:. The interaction between the modes is determined by the mode interaction coefficient. Since is symmetric and when , it is more efficient to use the equivalent expression. This expression is actually valid for several evaluation rules.
The only difference is how is computed. Several such expressions are given below. When a method is referred to as CQC, it is usually implied that the Der Kiureghian correlation coefficient is used. Here, and are the natural frequencies of the two modes, and and are the corresponding modal damping ratios.
It is possible that the response from two different modes, and , have different signs, so that a cross term can give a negative contribution to the sum. This is intentional, but it is a common misconception that the absolute values of and should be used. However, the underlying analysis contains an assumption about the response being a linear function of the mode shape.
If this it not the case, using absolute values is a safer approach. The most common nonlinear result quantities, like effective stresses, are always positive, in which case all terms in the sum will give a positive contribution anyway.
The strength of the correlation between two modes depends on the frequency ratio for the modes, but it also strongly depends on the damping. The double sum method uses a mode interaction coefficient , which is called the Rosenblueth correlation coefficient. It is conceptually similar to the Der Kiureghian correlation coefficient.
NRC Regulatory Guide 1. The damping coefficients are assumed to be constant outside the range of specified frequencies. Figure 1 illustrates how the damping coefficients at different eigenfrequencies are determined for the following input:. Mode selection and modal damping must be specified in the same way, using either mode numbers or a frequency range.
If you do not select any modes, all modes extracted in the prior frequency analysis, including residual modes if they were activated, will be used in the superposition. If you do not specify damping coefficients for modes that you have selected, zero damping values will be used for these modes. Damping coefficients for selected modes that are beyond the specified frequency range are constant and equal to the damping coefficient specified for the first or the last frequency depending which one is closer.
This is consistent with the way Abaqus interprets amplitude definitions. All points constrained by boundary conditions and the ground nodes of connector elements are assumed to move in phase in any one direction. This base motion can use a different input spectrum in each of three orthogonal directions two directions in a two-dimensional model.
Secondary bases cannot be used in a response spectrum analysis. No other loads can be prescribed in a response spectrum analysis. The density of the material must be defined Density. The following material properties are not active during a response spectrum analysis: plasticity and other inelastic effects, rate-dependent material properties, thermal properties, mass diffusion properties, electrical properties, and pore fluid flow properties—see General and perturbation procedures.
The value of an output variable such as strain, E; stress, S; or displacement, U, is its peak magnitude. Neither element energy densities such as the elastic strain energy density, SENER nor whole element energies such as the total kinetic energy of an element, ELKE are available for output in response spectrum analysis.
Reaction force output is not supported for response spectrum analysis using eigenmodes extracted using a SIM -based frequency extraction procedure with either the AMS or Lanczos eigensolver. Reaction force output in response spectrum analysis using eigenmodes extracted with the default Lanczos eigensolver provides directional combinations of so-called, modal reaction forces weighted with maximal absolute values of corresponding generalized displacements.
Directional and modal combination rules used for the reaction force calculation are the same as for other nodal output variables. Modal reaction forces are calculated in the frequency extraction procedure.
They represent static reaction forces calculated for the normal mode shapes. Generally, they cannot adequately represent reaction force in dynamic analysis. For models with diagonal mass and diagonal damping matrices the superposition of the modal reaction forces can provide a reasonable approximation of a nodal reaction force in mode-based analyses other than response spectrum analysis.
In response spectrum analysis the model response can be better represented by requesting section stresses and section forces in structural elements containing supported nodes. Response spectrum analysis Response spectrum analysis can be used to estimate the peak response displacement, stress, etc. Specifying a spectrum The response spectrum method is based on first finding the peak response to each base motion excitation of a one degree of freedom system that has a natural frequency equal to the frequency of interest.
Defining a spectrum using values of S as a function of frequency and damping You can define a spectrum by specifying values for the magnitude of the spectrum; frequency, in cycles per time, at which the magnitude is used; and associated damping, given as a ratio of critical damping.
Specifying the type of spectrum You can indicate whether a displacement, velocity, or acceleration spectrum is given. Creating a spectrum from a given time history record If you have a time history of a dynamic event e. Specifying the type of spectrum to be created You can indicate whether a displacement, velocity, or acceleration spectrum is to be created. Specifying the record type that the time history represents You can indicate whether a displacement, velocity, or acceleration amplitude is specified.
Creating an absolute or relative acceleration spectrum When you create an acceleration spectrum from a given time history record, you can create an absolute or relative response spectrum. Generating the list of damping values for the fraction of critical damping You must provide a list of damping values for the fraction of critical damping to create a spectrum.
Writing the generated spectra to an independent file You can write the generated spectra to an independent file. Combining the individual peak responses The individual peak responses to the excitations in different directions will occur at different times and, therefore, must be combined into an overall peak response.
Two combinations must be performed, and both introduce approximations into the results: The multidirectional excitations must be combined into one overall response. Directional summation methods You choose the method for combining the multidirectional excitations depending on the nature of the excitations.
The square root of the sum of the squares directional summation method If the spectra in different directions represent independent excitations, the modal summation is performed first, as explained below in Modal summation methods.
The forty-percent method If the spectra in different directions represent independent excitations, the modal summation is performed first, as explained below in Modal summation methods.
The thirty-percent method If the spectra in different directions represent independent excitations, the modal summation is performed first, as explained below in Modal summation methods. Modal summation methods The peak response of some physical variable R i a component i of displacement, stress, section force, reaction force, etc. The absolute value method The absolute value method is the most conservative method for combining the modal responses.
The square root of the sum of the squares modal summation method The square root of the sum of the squares method is less conservative than the absolute value method. The Naval Research Laboratory method The absolute value and square root of the sum of the squares methods can be combined to yield the Naval Research Laboratory method. The ten-percent method The ten-percent method recommended by Regulatory Guide 1. The complete quadratic combination method Like the ten-percent method, the complete quadratic combination method improves the estimation for structures with closely spaced eigenvalues.
The grouping method This method, also known as the NRC grouping method, improves the response estimation for structures with closely spaced eigenvalues. Double sum combination This method, also known as Rosenblueth's double sum combination Rosenblueth and Elorduy, , is the first attempt to evaluate modal correlation based on random vibration theory. Separation of modal responses into periodic and rigid responses Each spectrum can be divided into low-, medium-, and high-frequency regions.
Selecting the modes and specifying damping You can select the modes to be used in modal superposition and specify damping values for all selected modes. Specifying damping Damping is almost always specified for a mode-based procedure; see Material damping. Example of specifying damping Figure 1 illustrates how the damping coefficients at different eigenfrequencies are determined for the following input: Figure 1.
Damping values specified by frequency range. Rules for selecting modes and specifying damping coefficients The following rules apply for selecting modes and specifying modal damping coefficients: No modal damping is included by default.
Damping is applied only to the modes that are selected. Initial conditions It is not appropriate to specify initial conditions in a response spectrum analysis. Boundary conditions All points constrained by boundary conditions and the ground nodes of connector elements are assumed to move in phase in any one direction.
Predefined fields Predefined fields, including temperature, cannot be used in response spectrum analysis. Material options The density of the material must be defined Density. GV Generalized velocities for all modes. Technical Knowledge Base. Page tree. Browse pages. A t tachments 0 Page History Scaffolding History. Copy with Scaffolding XML.
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